Hey there, @slim 
Thanks so much for following up! My apologies for the delay here.
Something I missed on my end is that the drawer is no longer draggable. This is my mistake for missing this when you first reached out. The announcement in the Forums is here: Deploy Previews: New Dock UI, Embed Figma and Notion files in comments, and Jira integration That being said, the last icon on the far right side allows you to collapse it, so you can collapse it as an interim work around or also dismiss it entirely until you refresh the page! I know this isn’t entirely what you asked for.
Regarding public previews, anyone with a link to the deploy preview can request to edit. That is outlined here: How to Use Collaborative Deploy Previews with Netlify. A workaround for this would be turning off the ability to request access, by disabling the “Netlify Drawer”: Netlify App.
Again, I have filed your initial feature request with the team and they are aware of your feedback!